Inspection and GD&T
V-groove measuring calculator
A 0.250 in pin in a 90° groove 0.500 in wide sits with its top 0.0518 in above the surface; the groove is 0.2500 in deep to the sharp vertex.
Technical review pending. Formulas and values are cited. This notice comes down after a machinist review.
Pin in a known groove
Pin top relative to the surface (h)
in
h = r x (1 + 1 / sin(theta)) - Hv = 0.125 x (1 + 1 / sin(45°)) - 0.2500 = +0.0518 in
Above the surface by 0.0518 in (1.315 mm). Pin center c = r / sin(theta) = 0.1768 in above the vertex; pin top 0.3018 in above the vertex.
Groove depth to the sharp vertex (Hv)
in
Hv = W / (2 x tan(theta)) = 0.5 / (2 x tan(45°)) = 0.2500 in
6.350 mm. Width 0.1000 in below the surface: Wy = W - 2 x y x tan(theta) = 0.3000 in.
Flush pin diameter (d0)
in
d0 = 2 x Hv x sin(theta) / (1 + sin(theta)) = 2 x 0.2500 x sin(45°) / (1 + sin(45°)) = 0.2071 in
Contact 0.0884 in above the vertex, below the 0.2500 in groove depth: the pin touches the flanks. Largest usable pin dmax 0.7071 in. Pin bottom sits 0.0518 in above the vertex, so it clears a bottom flat narrower than 0.1036 in.
Source: Machinery's Handbook (measuring V-grooves and dovetails with pins); derived by right-triangle trigonometry as shown.
Metric results are shown to 0.001 mm because pin measurements are read that fine.
Results assume a true V with straight flanks and sharp top edges. Compare the reading with the drawing and its tolerance.
How it works
Hv = W / (2 x tan(theta)); c = r / sin(theta); h = r x (1 + 1 / sin(theta)) - Hv; d0 = 2 x Hv x sin(theta) / (1 + sin(theta)); dmax = 2 x Hv x sin(theta) / cos^2(theta); Wy = W - 2 x y x tan(theta); two pins: Delta = h2 - h1; sin(theta) = (d2 - d1) / (2 x Delta - (d2 - d1)); Hv = (d1 / 2) x (1 + 1 / sin(theta)) - h1
| Symbol | Meaning | Unit |
|---|---|---|
| A | Included groove angle | ° |
| theta | Half angle, A / 2 | ° |
| W | Groove width at the top surface, between the sharp edges | in or mm |
| d, r | Pin or ball diameter and radius | in or mm |
| Hv | Depth from the surface to the sharp vertex | in or mm |
| c | Pin center height above the vertex | in or mm |
| h | Pin top height relative to the surface (+ above, - below) | in or mm |
| d0 | Pin diameter that sits flush with the surface | in or mm |
| dmax | Largest pin that still touches the flanks | in or mm |
| y, Wy | Depth below the surface and the groove width there | in or mm |
| F | Bottom flat width | in or mm |
| d1, d2 | Smaller and larger pin or ball diameter | in or mm |
| h1, h2 | Pin top readings for d1 and d2 (+ above, - below the surface) | in or mm |
| Delta | h2 - h1 | in or mm |
Where the formula comes from
- The groove flanks meet at the sharp vertex. Half the width over the depth is tan(theta), so Hv = W / (2 x tan(theta)).
- The pin touches both flanks, so its center sits on the groove's center line. The radius to each contact point is square to the flank.
- That radius is the side opposite theta in a right triangle whose long side runs from the vertex to the pin center: c = r / sin(theta).
- The pin top is one more radius up: r x (1 + 1 / sin(theta)) above the vertex. Subtract Hv to read it from the surface: h = r x (1 + 1 / sin(theta)) - Hv.
- The contact point sits r x cos^2(theta) / sin(theta) above the vertex. It has to stay at or below the top edge, Hv, which gives the largest pin, dmax = 2 x Hv x sin(theta) / cos^2(theta).
- Set h = 0 and solve for d to get the flush pin: d0 = 2 x Hv x sin(theta) / (1 + sin(theta)).
- The pin bottom sits r x (1 / sin(theta) - 1) above the vertex. A bottom flat of width F sits F / (2 x tan(theta)) above it, so the pin clears the flat while its bottom is higher than that.
Groove angle from two pins
Included groove angle (A)
°
sin(theta) = (d2 - d1) / (2 x Delta - (d2 - d1)) = (6 - 4) / (2 x 3.000 - (6 - 4)) = 0.5000
theta = asin(0.5000) = 30.0000°; A = 2 x theta = 60.0000° (60° 0' 0"). Delta = h2 - h1 = -1.392 - (-4.392) = 3.000 mm.
Groove depth to the sharp vertex (Hv)
mm
Hv = (d1 / 2) x (1 + 1 / sin(theta)) - h1 = (4 / 2) x (1 + 1 / sin(30°)) - (-4.392) = 10.392 mm
Top width W = 2 x Hv x tan(theta) = 12.000 mm.
Worked example: 0.250 in pin in a 90° groove
90° groove, W = 0.500 in, 0.250 in pin (r = 0.125 in).
- theta = 45°; sin 45° = 0.70711; tan 45° = 1.0000.
- Depth: Hv = 0.500 / (2 x 1.0000) = 0.2500 in.
- Pin center: c = 0.125 / 0.70711 = 0.1768 in (0.17678 unrounded); pin top above the vertex = 0.17678 + 0.125 = 0.3018 in (0.125 x 2.41421).
- Reading: h = 0.3018 - 0.2500 = +0.0518 in above the surface (0.05178 unrounded; 1.315 mm).
- Contact height = 0.125 x 0.5 / 0.70711 = 0.0884 in, below 0.2500 in, so the pin touches the flanks.
- Flush pin: d0 = 2 x 0.2500 x 0.70711 / 1.70711 = 0.2071 in. Largest pin: dmax = 2 x 0.2500 x 0.70711 / 0.5 = 0.7071 in.
- Width 0.100 in below the surface: 0.500 - 2 x 0.100 x 1.0000 = 0.3000 in.
- Pin bottom above the vertex = 0.125 x (1.41421 - 1) = 0.0518 in, so it clears a bottom flat narrower than 2 x 0.0518 x 1.0000 = 0.1036 in.
Result: the pin top reads +0.0518 in above the surface; the groove is 0.2500 in deep.
Worked example: 6 mm ball in a 60° groove, then the angle from two balls
60° groove, W = 12 mm, 6 mm ball (r = 3 mm).
- theta = 30°; sin 30° = 0.5; tan 30° = 0.57735; cos 30° = 0.86603.
- Depth: Hv = 12 / (2 x 0.57735) = 10.392 mm.
- Ball center: c = 3 / 0.5 = 6.000 mm; top above the vertex = 6.000 + 3 = 9.000 mm.
- Reading: h = 9.000 - 10.392 = -1.392 mm: the ball top sits 1.392 mm below the surface (0.0548 in).
- Contact height = 3 x 0.75 / 0.5 = 4.500 mm, below 10.392 mm. Flush ball: d0 = 2 x 10.392 x 0.5 / 1.5 = 6.928 mm. Largest ball: dmax = 2 x 10.392 x 0.5 / 0.75 = 13.856 mm.
- Width 4 mm below the surface: 12 - 2 x 4 x 0.57735 = 7.381 mm.
- Angle from two balls: a 4 mm ball reads 4.392 mm below the surface (h1 = -4.392 mm) and the 6 mm ball reads 1.392 mm below (h2 = -1.392 mm). Delta = -1.392 - (-4.392) = 3.000 mm.
- sin(theta) = (6 - 4) / (2 x 3.000 - 2) = 2 / 4.000 = 0.5000; theta = 30.0000°; A = 60.0000°.
- Hv = 2 x (1 + 1 / 0.5) - (-4.392) = 6.000 + 4.392 = 10.392 mm; W = 2 x 10.392 x 0.57735 = 12.000 mm (11.9996 unrounded).
Result: the 6 mm ball reads -1.392 mm (below the surface); two balls give A = 60.0000°, Hv 10.392 mm, W 12.000 mm.
Shop notes
- Read h with a depth micrometer, or a height gage and indicator, from the top surface to the top of the pin.
- The formulas assume sharp groove edges and a straight flank. Broken or chamfered edges change W, not the pin reading; measure W at the sharp-edge intersection or work from Hv.
- For the angle method, use the biggest diameter difference the groove allows. With 0.1875 in and 0.2500 in pins in a 90° groove, a 0.0001 in error in one reading moves the angle about 0.26° (computed); with 4 mm and 6 mm balls in a 60° groove, a 0.001 mm error moves it about 0.03°.
- Use gage pins or balls of known size only; a drill shank is not round enough to measure with.
FAQ
Where does a pin sit in a 90° V-groove?
Its center sits r / sin(45°) = 1.4142 x r above the vertex. A 0.250 in pin in a 0.500 in wide groove tops out 0.0518 in above the surface.
What size pin sits flush in a V-groove?
d0 = 2 x Hv x sin(A/2) / (1 + sin(A/2)). For a 90° groove 0.2500 in deep, 0.2071 in.
How deep is a 60° V-groove 12 mm wide?
10.392 mm to the sharp vertex: 12 / (2 x tan 30°).
How do you find a V-groove angle with two pins?
Read both pin tops from the surface, then sin(A/2) = (d2 - d1) / (2 x Delta - (d2 - d1)). Readings 3.000 mm apart with 4 mm and 6 mm balls give 60°.
How big a pin can measure a V-groove?
Up to dmax = 2 x Hv x sin(A/2) / cos^2(A/2), when the contact reaches the top edge: 0.7071 in for a 90° groove 0.2500 in deep. Use a pin well under that.
Does a V-groove need a sharp bottom?
No. The pin touches only the flanks; a 0.250 in pin in a 90° groove clears a bottom flat up to 0.1036 in wide.
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